Generalised Beauville groups

A Beauville surface is defined by the action of a finite group (Beauville group) on the product of two compact Riemann surfaces. In this paper, we consider higher products and the possibility of a similar action by a finite group, which we call a generalised Beauville group; we prove several results regarding the existence and construction of infinite families of generalised Beauville groups and provide a complete classification of the abelian ones; we list all generalised Beauville groups of orders from 1 to 1023.

Weierstrass points on modular curves X0(N) fixed by the Atkin–Lehner involutions

PurposeThe authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ on X0(N) for N ≤ 50 are Weierstrass points or not.Design/methodology/approachThe design is by using Lawittes's and Schoeneberg's theorems.FindingsFinding all Weierstrass points on X0(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.Originality/valueThe Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X0(N).

Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

The category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface $$X_C$$ X C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface $$X_C$$ X C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface $$X_C$$ X C is called symmetric and the curve C is real.

Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus g in the base. These non-flat fibers contribute Kähler moduli to the four-fold but also add to the three-form cohomology for g > 0. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional $$ \mathcal{N} $$ N = (1, 0) superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively the non-flat fibers can be avoided by performing birational base changes analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.

Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

We consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

Complete set of Genera of Compact Riemann Surfaces with reference to the point Group of Sulphur (S8) Molecule

In this paper we consider the group of symmetries of the Sulphur molecule (S8 ) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u\upsilon/u2= \upsilon8 = (u\upsilon)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which D16 acts as a group of automorphisms as follows: D16 the group of symmetries of the sulphur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers \lambda and \mu such that \lambda \leq 1 and \mu \geq 1 and g=\lambda +8\mu (\geq2) , \mu\geq |\lambda|